Elliptic and Parabolic Equations - Abstract
Mielke, Alexander
Rate-independent systems can be understood as generalized gradient flows that are formulated in terms of a time-dependent energy functional $mathcal E$ and a dissipation potential $mathcal R$ in the form [ 0 in partial_dot z mathcal R(z,dot z) +mathrm D mathcal E(t,z). ] Applications occur in dry friction, elastoplasticity, magnetism, and phase-transforming systems like shape-memory alloys. The unique feature is that $mathcal R$ is homogeneous of degree 1 in the rate $dot z$, while it is quadratic in viscous cases leading to classical parabolic systems. This makes the relation between the rate $dot z$ and the dissipation force $partial_dot z mathcal R(z,dot z)$ homogeneous of degree 0 and hence discontinuous.
Since rate-independence occurs as a limit for systems under very slow loading, solutions may develop jumps when forced to leave a potential well. To model the correct jump behavior, we study the `limit of vanishing viscosity', where $mathcal R$ is replaced by [ mathcal R_varepsilon(z,v)=mathcal R(z,v)+fracvarepsilon2 langle mathbb V v,vrangle. ] The limit $varepsilon to 0$ leads to a new type of solutions for rate-independent systems, namely `parametrized solutions' and `BV solutions'. Deriving new a `rate-independent' priori estimate, we obtain convergence results for the vanishing-viscosity limit as well as for suitable time-incremental approximations, where viscosity and the time-steps tend to 0 simultaneously.
The work was done jointly with Riccarda Rossi, Giuseppe Savaré, and Sergey Zelik.